Discontinuous Galerkin Methods for Incompressible Flow

Discontinuous Galerkin (DG) discretizations for the three parts of the Navier-Stokes equations are discussed. First, we show the very natural derivation of the DG method by LeSaint and Raviart for advection problems. Then, the LDG method for the Laplacian and the Stokes operator is presented. It is combined with the method for advection to obtain a stable discretization of the Oseen equations. Finally, this method is applied inside a nonlinear iterative scheme to obtain strongly divergence free solutions of the incompressible Navier-Stokes equations. In the second part, the construction of efficient solvers for the discrete problems is discussed.